Concept

# Residual (numerical analysis)

Summary
Loosely speaking, a residual is the error in a result. To be precise, suppose we want to find x such that : f(x)=b. Given an approximation x0 of x, the residual is : b - f(x_0) that is, "what is left of the right hand side" after subtracting f(x0)" (thus, the name "residual": what is left, the rest). On the other hand, the error is : x - x_0 If the exact value of x is not known, the residual can be computed, whereas the error cannot. Residual of the approximation of a function Similar terminology is used dealing with differential, integral and functional equations. For the approximation f_\text{a} of the solution f of the equation : T(f)(x)=g(x) , , the residual can either be the function : ~g(x)~ - ~T(f_\text{a})(x), or can be said to be the maximum of the norm of this difference : \max_{x\in \mathcal X} |g(x)-T(f_\text{a})(x)| over the domain \mathcal X
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